In the case where \({\Omega }\)  is a planar Jordan \(\left(W^{1, p}, W^{1,p}\right)\) -extension domain, \({\Omega }\)  has to be a so-called John domain when \(1\le p\le 2\)  and the complementary domain has to be John when \(2\le p<\infty \) . The John condition implies that the Hausdorff dimension of \(\partial {\Omega }\)  must be strictly less than 2, see [1]}. Recently, Lu\(\check{c}\) ić, Takanen and the first named author gave a sharp estimate on the Hausdorff dimension of \(\partial {\Omega }\) , see [2]}. In general, the Hausdorff dimension of a \((W^{1, p}, W^{1, p})\) -extension domain can well be \(n\) .
